%I M1072 N0405 #61 Jan 15 2024 12:02:23

%S 1,1,2,4,7,13,24,43,78,141,253,456,820,1472,2645,4749,8523,15299,

%T 27456,49267,88407,158630,284622,510683,916271,1643963,2949570,

%U 5292027,9494758,17035112,30563634,54835835,98383803,176515310,316694823,568197628,1019430782

%N Number of partitions of n into parts 1/2, 3/4, 7/8, 15/16, etc.

%C Row sums of A049286 and A047913. [_Vladeta Jovovic_, Dec 02 2009]

%C Also number of compositions (a_1,a_2,...) of n with each a_i <= 2*a_(i-1). [_Vladeta Jovovic_, Dec 02 2009]

%D Minc, H.; A problem in partitions: Enumeration of elements of a given degree in the free commutative entropic cyclic groupoid. Proc. Edinburgh Math. Soc. (2) 11 1958/1959 223-224.

%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H Alois P. Heinz, <a href="/A002843/b002843.txt">Table of n, a(n) for n = 0..2000</a> (first 201 terms from Vincenzo Librandi)

%H David Benson, Pavel Etingof, <a href="https://arxiv.org/abs/2008.13149">On cohomology in symmetric tensor categories in prime characteristic</a>, arXiv:2008.13149 [math.RT], 2020.

%H R. K. Guy, Letter to N. J. A. Sloane, June 24 1971: <a href="/A002572/a002572.jpg">front</a>, <a href="/A002572/a002572_1.jpg">back</a> [Annotated scanned copy, with permission]

%H H. Minc, <a href="http://dx.doi.org/10.1017/S0013091500021945">A problem in partitions: Enumeration of elements of a given degree in the free commutative entropic cyclic groupoid</a>, Proc. Edinburgh Math. Soc. (2) 11 1958/1959 223-224.

%H Simon Plouffe, <a href="https://arxiv.org/abs/0911.4975">Approximations de séries génératrices et quelques conjectures</a>, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.

%H Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992

%F The g.f. (z**2+z+1)*(z-1)**2/(1-2*z-z**3+3*z**4) conjectured by _Simon Plouffe_ in his 1992 dissertation is wrong.

%e A straightforward partition problem: 1 = 1/2 + 1/2 and there is no other partition of 1, so a(1)=1.

%e a(3)=4 since 3 = 6(1/2) = 4(3/4) = 2(3/4) + 3(1/2) = 2(7/8) + 3/4 + 1/2.

%e a(4)=7 since 4 = 8(1/2) = 5(1/2) + 2(3/4) = 2(1/2) + 4(3/4) = 3(1/2) + 3/4 + 2(7/8) = 3(3/4) + 2(7/8) = 1/2 + 4(7/8) = 2(15/16) + 7/8 + 3/4 + 1/2.

%e From _Joerg Arndt_, Dec 28 2012: (Start)

%e There are a(6)=24 compositions of 6 where part(k) <= 2 * part(k-1):

%e [ 1] [ 1 1 1 1 1 1 ]

%e [ 2] [ 1 1 1 1 2 ]

%e [ 3] [ 1 1 1 2 1 ]

%e [ 4] [ 1 1 2 1 1 ]

%e [ 5] [ 1 1 2 2 ]

%e [ 6] [ 1 2 1 1 1 ]

%e [ 7] [ 1 2 1 2 ]

%e [ 8] [ 1 2 2 1 ]

%e [ 9] [ 1 2 3 ]

%e [10] [ 2 1 1 1 1 ]

%e [11] [ 2 1 1 2 ]

%e [12] [ 2 1 2 1 ]

%e [13] [ 2 2 1 1 ]

%e [14] [ 2 2 2 ]

%e [15] [ 2 3 1 ]

%e [16] [ 2 4 ]

%e [17] [ 3 1 1 1 ]

%e [18] [ 3 1 2 ]

%e [19] [ 3 2 1 ]

%e [20] [ 3 3 ]

%e [21] [ 4 1 1 ]

%e [22] [ 4 2 ]

%e [23] [ 5 1 ]

%e [24] [ 6 ]

%e (End)

%p b:= proc(n, i) option remember; `if`(n=0, 1,

%p add(b(n-j, min(n-j, 2*j)), j=1..i))

%p end:

%p a:= n-> b(n$2):

%p seq(a(n), n=0..40); # _Alois P. Heinz_, Jun 24 2017

%t v[c_, d_] := v[c, d] = If[d < 0 || c < 0, 0, If[d == c, 1, Sum[v[i, d - c], {i, 1, 2*c}]]]; Join[{1}, Plus @@@ Table[v[d, c], {c, 1, 34}, {d, 1, c}]] (* _Jean-François Alcover_, Dec 10 2012, after _Vladeta Jovovic_ *)

%Y Cf. A047913, A049286.

%K nonn,nice

%O 0,3

%A _N. J. A. Sloane_

%E More terms from _John W. Layman_, Nov 24 2001

%E Examples and offset corrected by Larry Reeves (larryr(AT)acm.org), Jan 06 2005

%E Further terms from _Vladeta Jovovic_, Mar 13 2006